Estimating Parameters under Equality and Inequality Restrictions

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Estimating Parameters under Equality and Inequality Restrictions

Adarsha Kumar Jena

Department of Mathematics, NITR

National Institute of Technology Rourkela

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Estimating Parameters under Equality and Inequality Restrictions

Dissertation submitted in partial fulfillment of the requirements of the degree of

Doctor of Philosophy

in

Mathematics

by

Adarsha Kumar Jena

(Roll Number: 512MA1006)

based on research carried out under the supervision of Prof. Manas Ranjan Tripathy

June, 2018

Department of Mathematics, NITR

National Institute of Technology Rourkela

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Department of Mathematics, NITR

National Institute of Technology Rourkela

June 01, 2018

Certificate of Examination

Roll Number: 512MA1006 Name: Adarsha Kumar Jena

Title of Dissertation: Estimating Parameters under Equality and Inequality Restrictions We the below signed, after checking the dissertation mentioned above and the official record book (s) of the student, hereby state our approval of the dissertation submitted in partial fulfillment of the requirements of the degree ofDoctor of PhilosophyinMathematicsatNational Institute of Technology Rourkela. We are satisfied with the volume, quality, correctness, and originality of the work.

Manas Ranjan Tripathy Kishor Chandra Pati

Principal Supervisor Member, DSC

Gopal Krishna Panda Durga Prasad Mohapatra

Member, DSC Member, DSC

Snehashish Chakraverty

External Examiner Chairperson, DSC

Kishor Chandra Pati Head of the Department

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Department of Mathematics, NITR

National Institute of Technology Rourkela

Prof. Manas Ranjan Tripathy Assistant Professor

June 01, 2018

Supervisor’s Certificate

This is to certify that the work presented in the dissertation entitled Estimating Parameters under Equality and Inequality Restrictions submitted by Adarsha Kumar Jena, Roll Number 512MA1006, is a record of original research carried out by him under my supervision and guidance in partial fulfillment of the requirements of the degree ofDoctor of Philosophy inMathematics. Neither this dissertation nor any part of it has been submitted earlier for any degree or diploma to any institute or university in India or abroad.

Manas Ranjan Tripathy

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Dedication

Dedicated to

my Parents

Adarsha Kumar Jena

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Declaration of Originality

I,Adarsha Kumar Jena, Roll Number512MA1006hereby declare that this dissertation entitled Estimating Parameters under Equality and Inequality Restrictions presents my original work carried out as a doctoral student of NIT Rourkela and, to the best of my knowledge, contains no material previously published or written by another person, nor any material presented by me for the award of any degree or diploma of NIT Rourkela or any other institution. Any contribution made to this research by others, with whom I have worked at NIT Rourkela or elsewhere, is explicitly acknowledged in the dissertation. Works of other authors cited in this dissertation have been duly acknowledged under the sections “Reference” or “Bibliography”.

I have also submitted my original research records to the scrutiny committee for evaluation of my dissertation.

I am fully aware that in case of any non-compliance detected in future, the Senate of NIT Rourkela may withdraw the degree awarded to me on the basis of the present dissertation.

June 01, 2018

NIT Rourkela Adarsha Kumar Jena

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Acknowledgment

Completion of this doctoral dissertation was possible with the support of several people. It is an honour and a great pleasure to express my sincere gratitude to all of them.

First and foremost, I would like to express my deep sense of gratitude and indebtedness to my supervisor Dr. Manas Ranjan Tripathy, Department of Mathematics, National Institute of Technology Rourkela for his effective guidance, lively discussion, constant encouragement, expert advice, constructive and honest criticism. Apart from the academic support, his friendly support helped me in many ways. While working with him, he made me realize my own strength and draw-backs, particularly boosted my self-confidence. I am also thankful to his family members for their continuous love, support and source of inspiration at all the time during my Ph.D. work.

I am grateful to Prof. Debasis Kundu, IIT Kanpur for his valuable suggestions during my research work. I am also grateful to Prof. Constance van Eeden, University of British Columbia, Canada and Prof. Hideki Nagatsuka, Tokyo Metropolitan University, Japan, for providing some valuable literature during my research work.

I would like to thank Director, National Institute of Technology Rourkela, for providing the facilities to carry out the research work. His leadership and management skills are always a source of inspiration. Moreover, I would like to thank my institute “National Institute of Technology, Rourkela” for providing me a vibrant research environment by conducting a number of seminars, workshops and conferences. I have enjoyed excellent computer facility, a clean and well furnished hostel, a well managed library and a green environment at my institute.

I am also grateful to all the members of my doctoral scrutiny committee Prof. K. C. Pati HOD, MA), Prof. S. Chakraverty, Prof. D. P. Mohapatra, Prof. G. K. Panda, National Institute of Technology Rourkela for their valuable suggestions, comments and timely evaluation of my research activity for the last five years. I am also thankful to all the faculty members, research scholars and office staffs of the Department of Mathematics, NIT Rourkela for their co-operation and encouragement during my tenure of PhD work.

Lastly, I am extremely grateful to my parents, sister and relatives who are a constant source of inspiration for me.

June 01, 2018 NIT Rourkela

Adarsha Kumar Jena Roll Number: 512MA1006

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Abstract

The problem of estimating statistical parameters under equality or inequality (order) restrictions has received considerable attention by several researchers due to its vast applications in various physical, industrial and biological experiments. For example, the problem of estimating the common mean of two normal populations when the variances are unknown has a long history and is popularly known as “common mean problem”. This problem is also referred as Meta-Analysis, where samples (data) from multiple sources are combined with a common objective. The “common mean problem” has its origin in the recovery of inter-block information when dealing with Balanced Incomplete Block Designs (BIBDs) problems. In this thesis, we study problem of estimating parameters and quantiles of two or more normal and exponential populations when the parameters are equal or ordered from decision theoretic point of view.

InChapter 1, we give the motivation and do a detailed review of literature for the following problems. InChapter 2, we discuss some basic definitions and decision theoretic results which are useful in developing the subsequent chapters. InChapter 3, the problem of estimating the common mean of two normal populations has been considered when it is known a priori that the variances are ordered. Under order restriction on the variances, some new alternative estimators have been proposed including one that uses the maximum likelihood estimator (MLE). These new estimators beat some of the existing popular estimators in terms of stochastic domination as well as Pitman measure of closeness criterion. In Chapter 4, we have considered the problem of estimating quantiles fork(≥2)normal populations with a common mean. A general result has been proved which helps in obtaining better estimators. Introducing the principle of invariance, sufficient conditions for improving estimators in certain equivariant classes have been derived. As a consequence some complete class results have been proved. A detailed simulation study has been carried out in order to numerically compare the performances of all the proposed estimators for the casesk = 3and4.A similar type of result has also been obtained for estimating the quantile vector. In Chapter 5, we deal with the problem of estimating quantiles and ordered scales of two exponential populations under equality assumption on the location parameters using type-II censored samples. First, we consider the estimation of quantiles of first population when type-II censored samples are available from two exponential populations. Sufficient conditions for improving equivariant estimators have been derived and as a consequence improved estimators have been obtained. A detailed simulation study has been carried out to compare the performances of improved estimators along with some of the existing ones. Further, we deal with the problem of estimating vector of ordered

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scale parameters. Under order restriction on the scale parameters, we derive the restricted maximum likelihood estimator for the vector parameter. We obtain classes of equivariant estimators and prove some inadmissibility results. Consequently, improved estimators have been derived. Finally a numerical comparison has been done among all the proposed estimators.

In Chapter 6, the problem of estimating ordered quantiles of two exponential populations is considered assuming equality of location parameters. Under order restriction, we propose new estimators which are the isotonized version of some baseline estimators. A sufficient condition for improving equivariant estimators are derived under order restriction on quantiles.

Consequently, estimators improving upon the baseline estimators are derived. Further, the problem of estimating ordered quantiles of two exponential populations is considered assuming equality of the scale parameters using type-II censored samples. Under order restrictions on the quantiles, isotonized version of some existing estimators have been proposed. Bayes estimators have been derived for the quantiles assuming order restriction on the quantiles. InChapter 7, we consider the estimation of the common scale parameter of two exponential populations when the location parameters satisfy a simple ordering. Bayes estimators using uniform prior and a conditional inverse gamma prior have been obtained. Finally all the derived estimators have been numerically compared along with some of the existing estimators. InChapter 8, we give an overall conclusion of the results obtained in the thesis and discuss some of our future research work.

Keywords: Admissibility; Bayes estimator; Common mean; Equivariant estimator;Inadmissibility;Isotonic regression;Maximum likelihood estimator(MLE);Ordered parameters;Quantiles;Quadratic loss;Relative risk performance;Squared error loss;Type-II censored samples;Uniformly minimum variance unbiased estimator(UMVUE).

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List of Figures

3.6.1(a)−(f)Comparison of risk values of several estimators for common meanµ using the lossL₁ for sample sizes(5,5),(12,12),(5,10),(10,5),(12,20)and (20,12)respectively. . . 57 4.3.1 Comparison of risk values of various estimators ofθ

e . . . 97 4.3.2 Comparison of risk values of various estimators ofθ

e . . . 98 5.2.1 Comparison of risk values of improved estimators for quantile θ whenk1 =

k2 = 0.5, m=n = 8andη= 0.01 . . . 123 5.2.2 Comparison of risk values of improved estimators for quantile θ whenk1 =

k2 = 1.0, m=n = 8andη= 1.50 . . . 124 5.3.1 Comparison of risk values of various estimators ofσ

e . . . 139 5.3.2 Comparison of risk values of various estimators ofσ

e . . . 140

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List of Tables

3.6.1 Percentage of risk improvements of all the proposed estimators using the loss L₁ for the sample sizes(m, n) = (5,5),(12,12),(20,20),(30,30) . . . 49 3.6.2 Percentage of risk improvements of all the proposed estimators using the loss

functionL₁for unequal sample sizes . . . 50 3.6.3 Percentage of risk improvements of all the proposed estimators using the L₂

andL₃loss functions . . . 51 3.6.4 Percentage of risk improvements of all the proposed estimators using the L₂

andL₃loss functions . . . 52 3.6.5 Percentage of risk improvements of all the proposed estimators using the L2

andL₃loss functions . . . 53 3.6.6 Percentage of relative risk improvements of all the proposed estimators using

L₁ loss function for equal sample sizes . . . 54 3.6.7 Percentage of relative risk improvements of all the proposed estimators using

theL₁loss function for unequal sample sizes . . . 55 4.2.1 Relative risk performances of various estimators for quantiles withη= 1.960 . 76 4.2.2 Relative risk performances of various estimators for quantiles withη= 1.960 . 77 4.2.3 Relative risk performances of various estimators for quantiles withη= 1.960 . 78 4.3.1 Percentage of relative risk improvements of various estimators of normal

quantiles withη= 1.960,(n1, n2) = (8,8),(12,12),(20,20),(40,40) . . . 94 4.3.2 Percentage of relative risk improvements of various estimators of normal

quantiles withη= 1.960,(n₁, n₂) = (4,10),(12,20),(30,40) . . . 95 4.3.3 Percentage of relative risk improvements of various estimators of normal

quantiles withη= 1.960, η = 1.960,(n₁, n₂) = (10,4),(20,12),(40,30) . . . 96 4.3.4 Relative risk performances of various estimators for quantile vector when k=3 . 99 4.3.5 Relative risk performances of various estimators for quantile vector when k=3 . 100 4.3.6 Relative risk performances of various estimators for quantile vector when k=3 . 101 4.3.7 Relative risk performances of various estimators for quantile vector when k=3 . 102 4.3.8 Relative risk performances of various estimators for quantile vector when k=3 . 103 4.3.9 Relative risk performances of various estimators for quantile vector when k=3 . 104

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5.2.1 Relative risk performances of different estimators for quantileθ whenη = 1.5 withk1 = k2 = 0.25,0.50,0.75,1.00 . . . 119 5.2.2 Relative risk performances of different estimators for quantileθ whenη = 1.5

withk1 = k2 = 0.25,0.50,0.75,1.00 . . . 120 5.2.3 Relative risk performances of different estimators for quantileθwhenη= 0.01

withk1 = k2 = 0.25,0.50,0.75,1.00 . . . 121 5.2.4 Relative risk performances of different estimators for quantileθwhenη= 0.01

withk1 = k2 = 0.25,0.50,0.75,1.00 . . . 122 5.3.1 Comparison of relative risk performances of different estimators ofσ

e= (σ₁, σ₂) when(m, n) = (8,8)andk1 =k2 = 0.25,0.50,0.75,1.00 . . . 141 5.3.2 Comparison of relative risk performances of different estimators ofσ

e= (σ₁, σ₂) when(m, n) = (12,12)andk1 =k2 = 0.25,0.50,0.75,1.00 . . . 142 5.3.3 Comparison of relative risk performances of different estimators ofσ

e= (σ₁, σ₂) when(m, n) = (12,20)andk₁ =k₂ = 0.25,0.50,0.75,1.00 . . . 143 5.3.4 Comparison of relative risk performances of different estimators ofσ

e= (σ₁, σ₂) when(m, n) = (20,12)andk₁ =k₂ = 0.25,0.50,0.75,1.00 . . . 144 6.2.1 Relative risk performance of various estimators for quantileθ₁

for(m, n) = (5,5),(10,10),(5,10),(10,5). . . 160 6.2.2 Relative risk performance of various estimators for quantileθ₂

for(m, n) = (5,5),(10,10),(5,10),(10,5). . . 161 6.3.1 Percentage of relative risk improvements of proposed estimators of θ₁ for

sample sizes (m, n) = (12,8) with censoring factors k1 = k2 = (0.25.0.5,0.75,1)and forη = 1.5;α= 3.5;β = 3.0 . . . 171 6.3.2 Percentage of relative risk improvements of proposed estimators of θ₁ for

sample sizes (m, n) = (8,12) with censoring factors k1 = k2 = (0.25.0.5,0.75,1)and forη = 1.5;α= 3.5;β = 3.0 . . . 172 6.3.3 Percentage of relative risk improvements of proposed estimators of θ₁ for

sample sizes (m, n) = (12,12) with censoring factors k1 = k2 = (0.25.0.5,0.75,1)and forη = 1.5;α= 3.5;β = 3.0 . . . 173 6.3.4 Percentage of relative risk improvements of proposed estimators of θ₂ for

sample sizes (m, n) = (12,8) with censoring factors k1 = k2 = (0.25.0.5,0.75,1)and forη = 1.5;α= 3.5;β = 3.0 . . . 174 6.3.5 Percentage of relative risk improvements of proposed estimators of θ₂ for

sample sizes (m, n) = (8,12) with censoring factors k1 = k2 = (0.25.0.5,0.75,1)and forη = 1.5;α= 3.5;β = 3.0 . . . 175 6.3.6 Percentage of relative risk improvements of proposed estimators of θ2 for

sample sizes (m, n) = (12,12) with censoring factors k1 = k2 = (0.25.0.5,0.75,1)and forη = 1.5;α= 3.5;β = 3.0 . . . 176

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7.4.1 Percentage of relative risk improvements of Bayes estimators

for sample sizes(m, n) = (4,5) . . . 188 7.4.2 Percentage of relative risk improvements of Bayes estimators

for sample sizes(m, n) = (25,25) . . . 191

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Chapter 1

Introduction

1.1 Introduction

The problem of estimating statistical parameters under equality or inequality (order) restrictions has received considerable attention by several researchers in the recent years. For example, the problem of estimating the common mean of two normal populations when the variances are unknown has a long history in the literature and is popularly known as “common mean problem”. This problem is also referred as Meta-Analysis, where samples (data) from multiple sources are combined with a common objective. The “common mean problem” has its origin in the recovery of inter-block information when dealing with Balanced Incomplete Block Designs (BIBDs) problems. Here two independent unbiased estimators (intra-block and inter-block) for the treatment contrasts are available. The target is to develop an estimator by combining intra-block and inter-block, which may perform better than either of these. Similarly, the problem of estimating parameters under certain inequality (order) restrictions is of considerable interest and has been extensively studied by several researchers in the recent past. This type of statistical models arise in various physical, agricultural, industrial, biological and medical experiments. Below we discuss certain practical situations where modeling of the problem leads to the assumption of equality or/and inequality restrictions on the involved parameters.

1. Suppose there are n laboratories or operators evaluating a given product. It is quite possible to assume that the locations of the measured aspect of the product to be the same, where as the scales may differ due to laboratory techniques or facilities. The assumption on the distribution of the measured quantity may follow a particular location-scale family.

2. A particular type of products (electrical/mechanical) has been manufactured by different companies and to be lunched in the market. Because of market restrictions, the minimum guarantee periods (location) of the products may be same, whereas the average lives (scale) may be different. The life times of the products may follow certain life-time distributions. On the basis of prior information, one may be interested to estimate the parameters.

3. Suppose the farmers of a country use three types of treatments to grow the crops:

treatment-I (using chemical fertilizers), treatment-II (using organic manures) and

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Chapter 1 Introduction treatment-III (without using any fertilizers). Let θ₁, θ₂, θ₃ be denote the average yields by using the three types of treatments respectively. It is natural that θ₁ ≥ θ₂ ≥ θ₃ and one would be interested in estimating one or all of(θ₁, θ₂, θ₃).

In this thesis, we have considered the problem of estimating equal or ordered parameters when the underlying distribution is either normal or exponential. Moreover, we have focused on estimating quantiles of these populations when the concerned or nuisance parameters are equal or ordered. We note that for these distributions, quantiles are linear function of location and scale parameters.

1.2 A Review of Literature

In this section we give a detailed review of literature on certain problems which are relevant and useful for developing the chapters of thesis.

1.2.1 Estimation of Common Mean of Two Normal Populations

The problem of estimating common mean of two normal populations is an age old problem and has a long history in the literature of statistical inference. The problem has received considerable attention by several authors in the last few decades due to its practical applications as well as theoretical challenges involve in it. Particularly, the problem has been well investigated from classical as well as decision theoretic point of view when there is no order restrictions on the variances. The problem is quite popular in the literature and is popularly known as

“common mean problem”. The problem has been originated from the study of recovery of inter-block information while dealing with balanced incomplete block design (BIBD) problems (see Shah ( 1964)). Probably, Yates (1940) was the first to consider the problem under normality assumption. Let(X_i1, X_i2, . . . , X_in_i);i = 1,2be a random sample taken from thei^th normal populationN(µ, σ²_i).The problem is to estimate the common parameterµwhen the variances are unknown and unequal with respect to the loss function

L₁(d, µ) = (d−µ)² or,

L₂(d, µ) =

(d−µ σ₁

)2

, wheredis an estimator forµ.Let us define the random variables

X¯_i = 1 n_i

ni

∑

j=1

X_ij, S_i² =

ni

∑

j=1

(X_ij −X¯_i)²;i= 1,2.

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Introduction Chapter 1 The random variables X¯_i and S_i² are statistically independent and minimal sufficient for this model. We also note that these minimal sufficient statistics are not complete.

One of the first pioneering research work in this direction was done by Graybill and Deal (1959). They proposed a new combined estimator for the common meanµby taking convex combination ofX¯1 andX¯2 with weights as the functions of sample variances. Their combined estimator is given by

dGD = n₁(n₁−1)S₂²X¯₁+n₂(n₂−1)S₁²X¯₂ n₁(n₁−1)S₂²+n₂(n₂−1)S₁² .

They have proved that the combined estimator d_GD performs better than bothX¯₁ and X¯₂ in terms of variances (loss function L₁) when the sample sizes are at least 11. The estimator d_GD is also known as the best asymptotically normal and conditionally unbiased. After then a lot of research work has been done in this direction by several authors using classical as well as decision theoretic approaches. Their target has been to derive either some alternative estimators forµwhich may compete withd_GD or proving some decision theoretic results like admissibility or minimaxity. Very surprisingly still now it remains an open problem whether d_GD is admissible or inadmissible.

For small sample sizes (n₁, n₂ ≤ 10) Zacks (1966) proposed two classes of testimators usingF-test. Let us denote τ = σ₂²/σ²₁.Case-1: Consider testing the hypothesisH0 : τ = 1 againstH1 :τ ̸= 1.IfH0is accepted then use the grand mean of two samples for estimatingµ, otherwise use the Graybill-Deal estimatord_GD.Mathematically the testimator is written as

d₁(τ^∗) = { _¯

X1+ ¯X2

2 , if _τ¹_∗ ≤ ^S_S²²2 1 ≤τ^∗ d_GD, otherwise,

where τ^∗ is the critical value of the F-tests and 1 ≤ τ^∗ ≤ ∞. Case-2: Consider the testing procedure for the three alternatives,H₀ : τ = 1, H₁ : τ >1andH₂ : τ <1.IfH₀ is true use the grand mean as an estimator forµ.If eitherH₁ orH₂ holds true then use the sample mean which has the smaller variance as the estimator forµ.This estimator can be written as

d₂(τ^∗) =









X¯₂, if ^S_S²²2 1 < _τ¹_∗

X¯1+ ¯X2

2 , if _τ¹_∗ ≤ ^S_S²²²

1 ≤τ^∗ X¯₁, if^S_S²²2

1

> τ^∗.

Finally the author compared numerically the performances of all the estimators in these two classes.

Mehta and Gurland (1969) considered the estimation of µ under the assumption that the nuisance parameters follow a certain simple ordering say σ²₁ ≤ σ²₂. They have proposed the following class of estimators.

d(ψ) =ψ(T) ¯X₁ + (1−ψ(T)) ¯X₂,

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Chapter 1 Introduction whereT = S₂²/S₁².Depending upon the choices ofψ,the following three types of estimators can be proposed.

(i) δ₁ =d(ψ),whereψ(T) = _c+a+T^(c+T⁾, (ii) δ₂ =d(ψ),where

ψ(T) = { 1

2, ifT < k

T

T+a, ifT ≥k.

(iii) δ₃ = d(ψ), where ψ(T) =

√(c+T)

√(c+T)+a and a, c, k are specific constants to be suitably chosen.

For the case n₁ = n₂, they proved that the estimator δ₁ performs better thand_GD for some choices ofa, c andk.Further for some choices ofk they have also proved that the estimator δ₂ performs better thand_GD.Finally, authors numerically compared the efficiencies of all the above three estimators.

Zacks (1970) considered the problem of estimation of common meanµusing the decision theoretic approach. The author proposed an equivariant class of estimators forµwhich is given by

dZ = ¯X1+ ( ¯X2−X¯1)ϕ(T1, T2), whereT₁ = _{( ¯}_X ^S¹

1−X¯2)²,andT₂ = _{( ¯}_X ^S²

2−X¯1)².This class contains estimators that was previously proposed by Zacks (1966) and Mehta and Gurland (1969). He also proved that the estimatorX¯₁ is minimax with respect to the lossL₂.Further using a symmetric loss function, he proved that the grand mean is minimax. Zacks also derived the generalized Bayes estimator with respect to the Jeffrey’s prior which is known as fiducial equivariant estimators. They also proved that these Bayes estimators are weakly admissible.

Khatri and Shah (1974) considered a general class of estimators forµwhich is given by dKS = (1−ϕ(W)) ¯X1+ϕ(W) ¯X2,

whereϕ(W) = _n ⁿ²^S¹²

2S₁²+n1cS₂² andc = ⁽ⁿ_(n¹⁻³⁾⁽ⁿ²⁻¹⁾

2−3)(n1−1).The estimatord_KS improves onX¯₁ in terms of variance ifn₂ ≥2.Furtherd_KS improves upon bothX¯₁ andX¯₂ if(n₁−7)(n₂−7) ≥16.

Hence the estimatord_KS can be used in certain situations whered_GD fails to improve uponX¯₁ andX¯₂.

Cohen and Sackrowitz (1974) constructed the following class of estimators whenn₁ =n₂ = n(say).

d_CS = (1−C_nH(z)) ¯X₁+C_nH(z) ¯X₂,

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Introduction Chapter 1 where

Cn=

{ _(n₋₃₎2

(n+1)(n−1), if n is odd,

n−4

n+2, if n is even.

and

H(z) =

{ F(1,(3−n)/2,(n−1)/2, z), for0≤z ≤1,

−_(n⁽ⁿ₋⁻_1)z³⁾F(1,(5−n)/2,(n+ 1)/2,1/z), forz ≥1.

Here F is the hyper geometric function. Forn ≥ 5,they have shown that the estimator d_CS improves uponX¯₁ usingL₁. Further for the casen ≥10,the estimator

d^∗_CS = (1−H(z)) ¯X₁ +H(z) ¯X₂ performs better than bothX¯₁ andX¯₂.

Brown and Cohen (1974) proposed the following class of estimators given by, d_BC1(b) = X¯₁+

{ (bS₁²/n₁(n₂−1))( ¯X₂−X¯₁)

S₁²/n₁(n₁+ 1) +S₂²/n₂(n₂+ 2) + ( ¯X₂−X¯₁)²/(n₂+ 2) }

, where 0 < b ≤ bmax(n1, n2) = 2(n2 + 2)/n2E(max(V⁻¹, V⁻²)). Here V follows F distribution with(n₂+2)and(n₁−1)degrees of freedom. The estimatord_BC1(b)is unbiased for µ.Whenn₂ ≥3,they have shown thatd_BC1(b)performs better thanX¯₁.They also established that forn₂ = 2, n₁ ≥2,the estimatord_BC1(b)is not better thanX¯₁for any choices ofb.Further, authors constructed a different class of unbiased estimators of the form

dBC2(p, b) = X¯1+

{ (bS₁²/n₁(n₁−1))( ¯X₂−X¯₁)

p(S₁²/n₁(n₁−1) +S₂²/n₂(n₂−1)) + (1−p)( ¯X₂−X¯₁)² }

, where 0 < p < 1and 0 < b < b_max(n₁, n₂ −3). Whenn₁ ≥ 2, n₂ ≥ 3, they have proved that there exist values ofb(>0)for whichdBC2(p, b)performs better thanX¯1.They have also generalized some of their results tok(≥2)normal populations.

Bhattacharya (1980) proposed a class of estimators that includes the estimators proposed by Brown and Cohen (1974) and Khatri and Shah (1974).

Sinha and Mouqadem (1982) proved the admissibility of the estimatord_GDin certain classes of estimators. They defined the classDand its membersD0, D1, D2 as follows.

D = {d= ¯X₁+ ( ¯X₂−X¯₁)ϕ; 0≤ϕ(S₁², S₂²,X¯₂−X¯₁)≤1}, D₀ = {d= ¯X₁+ ( ¯X₂−X¯₁)ϕ, 0≤ϕ(S₂²

S₁²)≤1}, D₁ = {d= ¯X₁+ ( ¯X₂−X¯₁)ϕ, 0≤ϕ(S₁², S₂²)≤1}, D₂ = {d= ¯X₁+ ( ¯X₂−X¯₁)ϕ, 0≤ϕ( S₁²

( ¯X2−X¯1)², S₂²

( ¯X2−X¯1)²)≤1}.

The loss function is taken asL₁.The authors proved that the estimatord_GDis admissible in the classD₀ forn₁ =n₂ ≥2and extended admissible inD.

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Chapter 1 Introduction Bhattacharya (1986) observed that the conclusions of Cohen and Sackrowitz (1974) regarding improvements upon X¯₁ and X¯₂ are not correct. He proved that the estimator d_CS dominatesX¯₁whenn≥7and, bothX¯₁ andX¯₂whenn≥15.

Kubokawa (1987a) considered a general class of estimators for estimatingµwhich is given by

d_ϕ(a, b, c) = ¯X₁+ a( ¯X₂−X¯₁)

1 +Rϕ(S₁², S₂²,( ¯X₂−X¯₁)²),

whereR={bS₂²+c( ¯X1−X¯2)²}/S₁²andϕis any positive real valued function. The estimator d_ϕ(a, b, c)improves uponX¯₁and is also minimax with respect to the lossL₂ when0≤a≤2, b≥c > 0.Furthermore, for the choice ofϕ= 1 +d/{bS₂²+c( ¯X₁−X¯₂)²},the above estimator is reduces to

d₁(a, b, c, d) = ¯X₁+ aS₁²( ¯X2−X¯1)

S₁²+bS₂²+c( ¯X₁−X¯₂) +d.

For particular choices ofa,b,candd,the above class produces estimators which were proposed by Graybill and Deal (1959), Khatri and Shah (1974), Brown and Cohen (1974), Bhattacharya (1979) and Kubokawa (1987b).

Kubokawa (1989) proposed a class of estimators that dominate X¯₁ in terms of Pitman Measure of Closeness (PMC). In particular he proved that the µˆ_GD dominates X¯₁ and X¯₂ if the sample sizes are at least5.

Nanayakkara and Cressie (1991) proposed a new class of estimators for the common mean µwhich is given by

d_{N C}(r) =

(α1X¯1

S₁^r +α2X¯2

S₂^r )

/ (α1

S₁^r + α2

S₂^r )

, r >0.

For the caser = 2,they have obtained necessary and sufficient condition on α₁ and α₂ for which the estimatord_{N C}(r)improves uponX¯₁andX¯₂.

Kelleher (1996) considered the problem of estimating common mean for small and equal sample sizes. The author obtained a Bayes estimator by considering a prior for the ratio of variancesσ₁²/σ²₂ =τ.

dB(R) =

∫_∞

0 τ f(R|τ)dQ(τ)

∫_∞

0 (τ+ 1)f(R|τ)dQ(τ).

whereR = S₂²/S₁². He also proved numerically thatd_B(R)perform better than the estimator proposed by Zacks (1966).

Mitra and Sinha (2007) studied the common mean problem from Bayesian point of view.

He has obtained the generalized Bayes estimator with respect to Jeffrey’s prior. The prior is

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Introduction Chapter 1 taken as

f(µ, σ²₁, σ₂²) = (

√

σ₁²+σ²₂)/(σ²₁σ₂²)^3/2, − ∞< µ <∞, σ₁² >0, σ²₂ >0.

The estimator is given by dM S =

∫_∞

0 τ^n/2(τ+ 1)ⁿ/(aτ²+bτ +c)ⁿ⁺¹dτ

∫_∞

0 τ^n/2(τ + 1)ⁿ⁺¹/(aτ²+bτ +c)ⁿ⁺¹dτ.

The authors also numerically compared the risk performance of dM S with the estimators proposed by Graybill and Deal (1959), Sinha (1979) and Sinha and Mouqadem (1982).

Pal et al. (2007) have obtained variance of the maximum likelihood estimator (MLE) ofµ and the variance ofd_GD numerically. They have shown by using simulation that, in most of the parameter ranges, the MLE has the smaller variance thand_GD.

Tripathy and Kumar (2010) revisited the problem of estimating common mean of two normal populations when the variances are unknown and unequal. Authors have established some decision theoretic results using a quadratic loss function. They have also obtained an alternative estimator forµ,by modifying the estimator proposed by Moore and Krishnamoorthy (1997). Their estimator is given by

dT K = X¯₁√

n₁c_n₂S₂+ ¯X₂√

n₂c_n₁S₁

√n₁c_n₂S₂+√

n₂c_n₁S₁ wherecni = ^Γ(

ni−1 2 )

√2Γ(ⁿⁱ₂ );i= 1,2.Authors also obtained sufficient conditions for improving certain classes of equivariant estimators for the common mean. Through a simulation study, they have numerically compared the risk values of all the proposed estimators and recommended for their use. Their numerical comparison reveals that the estimatord_{T K}compete well with the estimator proposed by Tripathy and Kumar (2010).

1.2.2 Estimating Common Mean of Several Normal Populations (A Generalization to k( ≥ 3) Populations)

In this section we review the literature on the problem of estimating common mean of k(≥ 3)normal populations when the variances are unknown and possibly unequal. Suppose (X_i1, X_i2, . . . , X_in_i);i= 1,2, . . . , k,be a random sample taken from thei^thnormal population N(µ, σ_i²).Consider the problem of estimatingµwith respect to the lossesL₁andL₂as defined in previous section. Let us define the random variables

X¯_i = 1 n_i

ni

∑

j=1

X_ij, S_i² =

ni

∑

j=1

(X_ij −X¯_i)², i= 1,2, . . . , k.

Probably Norwood and Hinkelmann (1977) was the first to consider the problem for(k ≥2) normal populations. In fact, the authors have generalized the estimator given by Graybill and

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Chapter 1 Introduction Deal (1959) tok(≥2)populations and the estimator is given by

d_{N H} =

∑k i=1

n_i(n_i−1) ¯X_i/S_i²

∑k i=1

n_i(n_i−1)/S_i² .

It has been shown that the estimatordN Hperforms better than each ofX¯iwith respect to the loss L₁ if and only if eachn_i ≥ 11or onen_i = 10,and all othern_j ≥ 18, i̸= j :j = 1,2, . . . , n_i andi= 1,2, . . . , k.

Shinozaki (1978) constructed a general class of estimators that contains d_{N H} and derived conditions for improving upon eachX¯_i. Their proposed estimator is given by

d_SZ =

∑k i=1

c_in_i(n_i−1) ¯X_i/S_i²

∑k i=1

c_in_i(n_i−1)/S_i² .

The author established that the estimatord_SZperforms better than eachX¯_iif and only ifn_i ≥8 and(ni−7)(nj−7)≥16fori̸=j.

Sinha (1979) established that the estimator d_{N H} is inadmissible with respect to a general type of loss function, when for somei σ_i² ≤ σ_j², i ̸= j.In fact Sinha’s result generalizes the inadmissibility result of Mehta and Gurland (1969).

Bhattacharya (1984) developed two general inequalities and used these to obtain a better estimator forµ.He also obtained improvements over shinozaki’s (Shinozaki (1978)) result.

Kubokawa (1987c) considered the estimation of µ for (k ≥ 2)normal populations with respect to a symmetric loss function defined by

L(d, µ) =ψ(|d−µ|^r), 0< r <∞,

whereψis a decreasing concave function of non negative real numbers and satisfiesψ(0) = 0.

Author proposed a general class of estimators which is given by d_K =

∑_k

i=1ci(ni−1) ¯Xi/S_i²

∑k

i=1c_i(n_i−1)/S_i² ,

wherec_is are positive constants. Further he proved that the estimatord_Kis better than eachX¯_i ifn_i ≥6andc_j/c_i ≤2(n_j −5)/(n_i+ 1)fori̸=j;i, j = 1,2, . . . , k.

Sarkar (1991) extended the results of kubokawa (Kubokawa (1989)) to k normal populations.

Moore and Krishnamoorthy (1997) constructed a new type of combined estimator forµby taking convex combination ofX¯_is with weights inversely proportinal to their standard errors

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Introduction Chapter 1 which is given by

d_{M K} =

∑_k

i=1

√n_i(n_i−1) ¯X_i/S_i

∑k i=1

√n_i(n_i−1)/S_i .

The authors numerically compared the estimatord_{M K}with that ofd_{N H}in terms of the variances through a simulation study. Their numerical study reveals that the estimator d_{M K} performs better thand_{N H} when either the sample sizes are small or the population variances are close to each other.

Tripathy and Kumar (2015) have investigated the problem of estimating common meanµof several normal populations using a decision theoretic approach with respect to a quadratic loss function. They have modified the estimator proposed by Moore and Krishnanmoorthy (1997) and is given by

d_{T K} =

∑_k

i=1

√n_iX¯_i/b_n_i₋₁S_i

∑k i=1

√n_i/b_n_i₋₁S_i .

The authors obtained classes of affine and location equivariant estimators and proved some inadmissibility results in these classes. As a consequence some complete class results have been proved. In addition to these, the authors also numerically compared the risk values of their proposed estimators with other well known estimators (including the MLE which has been obtained numerically) for the casek = 3,4using the Monte-Carlo simulation method. Finally recommendations have been made for the use of all these estimators for various choices of the parameters.

1.2.3 Estimating Common Mean (Variance) when the Nuisance Parameters are Ordered

The problem of estimating common mean or variance when the nuisance parameters (parameters other than our study of interest) satisfying certain ordering has received attentions by few researchers in the recent past. Suppose (X_i1, X_i2, . . . , X_in_i);i = 1,2 is a random sample taken from thei^th normal populationN(µ, σ_i²).The problem of interest is to estimate the common parameterµunder the assumption thatσ₁² ≤σ₂².

Perhaps Sinha (1979) was the first to consider this model with equal sample sizes when the loss function is strictly increasing in |d−µ|. He proposed a new estimator which dominates d_GD stochastically as well as universally. The proposed estimator is given by

d_S = {

d_GD, if _n^S¹²

1−1 ≤ _n^S₂₋²²₁

X¯1+ ¯X2

2 , if _n^S¹²

1−1 > _n^S²²

2−1.

Elfessi and Pal (1992) considered the same model and proposed new estimators for both

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Chapter 1 Introduction equal and unequal sample sizes. Their estimators are given by

dEP = {

d_GD, ifS₁² ≤S₂² δX¯₁+ (1−δ) ¯X₂, ifS₁² > S₂², and

d_EP = {

d_GD, if _n^S²¹

1−1 ≤ _n^S₂₋²²₁

n1X¯1+n2X¯2

n1+n2 , if _n^S²¹

1−1 > _n^S²²

2−1,

for equal and unequal sample sizes respectively, whereδ =S₁²/(S₁²+S₂²).The authors proved that the estimatordEP dominatesdGDuniversally as well as stochastically whenσ₁² ≤σ²₂.They also obtained the percentage of risk improvement of d_EP overd_GD using both absolute error loss and squared error loss, numerically.

Misra and van der Meulen (1997) generalized the results obtained by Elfessi and Pal (1992) tok(≥2)normal populations. Furthermore they have proved that the proposed new estimator dominates its old counter part (d_{N H},extension of Graybill-Deal estimator to the casek(≥ 2)) in terms of Pitman measure of closeness criteria.

Chang et al. (2012) considered the problem of estimating common and ordered means of two normal populations assuming that the variances follow a simple ordering. They have proposed a class of estimators for the common meanµof the form

ˆ

µ(γ) =γX¯₁+ (1−γ) ¯X₂,

whereγis a function ofs²₁, s²₂,x¯₁−x¯₂ and0≤γ ≤1.Herex¯_iands²_i are the sample mean and variance of thei^thpopulation respectively. They have proved that the estimatorµ(γ)ˆ dominates the Graybill-Deal estimator stochastically. Similarly they have chosen two classes of plug-in type estimators for the ordered meansµ1,andµ2;µ1 ≤µ2 as

ˆ

µ₁(γ) = min( ¯X1, γX¯₁+ (1−γ) ¯X₂) and µˆ₂(γ) =max( ¯X₁, γX¯₁+ (1−γ) ¯X₂),

respectively. The estimatorµˆ₂(γ)dominates stochasticallyX¯₂where as a similar type of result does not hold true in the case ofµˆ₁(γ).

Gupta and Singh (1992) investigated the problem of estimating common variance of two normal populations when it is known a priori that the means follow a simple ordering say µ₁ ≤µ₂.Under order restrictions on the means, authors established that the restricted MLEs of the common variance and the ordered means dominate their old counter parts (the unrestricted MLEs, that is, estimators without taking account order restrictions on the means) in terms of Pitman measure of closeness criteria.

Tripathy et al. (2013) considered the problem of estimating common standard deviation(σ) of two normal populations under order restrictions on the means using a scale invariant loss function. A general minimaxity result has been proved and a class of minimax estimators is derived. An admissibility result is proved in this class. Further a class of equivariant estimators

Estimating Parameters under Equality and Inequality Restrictions

Estimating Parameters under Equality and Inequality Restrictions

Adarsha Kumar Jena

Department of Mathematics, NITR

National Institute of Technology Rourkela

Estimating Parameters under Equality and Inequality Restrictions

Doctor of Philosophy

Mathematics

Adarsha Kumar Jena

Department of Mathematics, NITR

National Institute of Technology Rourkela

Department of Mathematics, NITR

National Institute of Technology Rourkela

Certificate of Examination

Department of Mathematics, NITR

National Institute of Technology Rourkela

Supervisor’s Certificate

Dedication

Dedicated to

my Parents

Declaration of Originality

Acknowledgment

Abstract

Contents

List of Figures

List of Tables

Chapter 1

Introduction

1.1 Introduction

1.2 A Review of Literature

1.2.1 Estimation of Common Mean of Two Normal Populations

1.2.2 Estimating Common Mean of Several Normal Populations (A Generalization to k( ≥ 3) Populations)

1.2.3 Estimating Common Mean (Variance) when the Nuisance Parameters are Ordered